\name{DeltaSD_lowerBound_Mpre}
\alias{DeltaSD_lowerBound_Mpre}
%- Also NEED an '\alias' for EACH other topic documented here.
\title{
Construct lower bound for M for Delta = Delta^{SD}(M) based on observed pre-period coefficients.
}
\description{
  Constructs a lower bound for M using the observed pre-period coefficients. It constructs a one-sided confidence interval for the maximal second difference of the observed pre-period using the conditional test developed in Andrews, Roth & Pakes (2019).
}
\usage{
DeltaSD_lowerBound_Mpre(betahat, sigma, numPrePeriods, alpha = 0.05, grid.ub = NA, gridPoints = 1000)
}
%- maybe also 'usage' for other objects documented here.
\arguments{
  \item{betahat}{
  Vector of estimated event study coefficients.
 }
 \item{sigma}{
  Covariance matrix of event study coefficients.
 }
 \item{numPrePeriods}{
  Number of pre-periods.
  }
  \item{alpha}{
  Desired size of the one-sided confidence set. Default equals 0.05 (corresponding to 95\% confidence interval)
  }
  \item{grid.ub}{
  Upper bound of grid of values of M that is used to construct the confidence interval by test inversion. Default equals NA and the upper bound of the grid is set equal to three times the maximum standard error of the observed pre-period event-study coefficients.
  }
  \item{gridPoints}{
  Number of points to include in the grid that is used to construct the confidence interval by test inversion. Default equals 1000 points.
  }
}
\details{
%%  ~~ If necessary, more details than the description above ~~
}
\value{
Returns a scalar that equals the lower bound of a one-sided confidence interval for the maximal second difference of the observed pre-period coefficients.
}
\references{
Andrews, Isaiah, Jonathan Roth and Ariel Pakes. "Inference for Linear Conditional Moment Inequalities." 2019.
Rambachan, Ashesh and Jonathan Roth. "An Honest Approach to Parallel Trends." 2019.
}
\author{
Ashesh Rambachan
}
\note{
%%  ~~further notes~~
}

%% ~Make other sections like Warning with \section{Warning }{....} ~

\seealso{
%% ~~objects to See Also as \code{\link{help}}, ~~~
}
% Add one or more standard keywords, see file 'KEYWORDS' in the
% R documentation directory.
\keyword{ ~kwd1 }% use one of  RShowDoc("KEYWORDS")
\keyword{ ~kwd2 }% __ONLY ONE__ keyword per line
